In the August/September 2003 issue of FOCUS, I wrote an article entitled "What's the Best Textbook — Elementary Number Theory," which I hoped would be the first of many such articles by many authors. (It didn't work.) In that article, I chose this book as the best textbook for (my version of) the Elementary Number Theory course, in part because this is "a book that works hard to be approachable." That is important for me, because the course I teach at Colby is intended to be accessible to a wide range of students. And, as I also explained, readability is important: I want my students to be able to read their textbook.

Of course, readability is not enough. The content must be right too. In an elementary book, I usually look for hints that the author understands the deeper parts of the subject. Most of my students are not yet able to decide by themselves what is deep and what is routine, what is a brilliant insight and what is just technical prowess. An author who really understands his subject can put up signposts to help students grasp such things. By doing so, he can also keep the book interesting for those students in my class who already have quite a bit of mathematical background.

On all of these measures, Silverman's book is excellent. It is insightful, interesting, and fun. From my article:

Silverman does a great job of exactly those things I care about: his book is readable; it makes students aware of the role of both experimentation and proof; it includes many pointers to deeper questions and even to open questions. Even the jokes are the sort that I would make!

What's more, my students seem to agree. They do not find the book easy, but they do seem to be able to read it and get something out of their reading.

Of course, no book is perfect. For me, this book's biggest fault is the fact that it doesn’t include a proof of quadratic reciprocity. This, of course, is easy to fix: I supply one on a handout, usually based on the one given in Dan Flath's Introduction to Number Theory.

Any author who calls his mathematics textbook "friendly" is being quite daring. Silverman pulls it off.

As far as I can tell, this new edition does not include all that many significant changes. Good old problem 21.3, which has always seemed very sneaky to me, is still there. So are all the neat chapters at the end, which I never seem to have time for. Maybe this year.